Can people actually see spells in the meat space world?
Which might end up breaking it down to the individual spell as I would believe that physical combat spells such as fireball would obviously be seen. Where as the manipulation spell influence would not leave any sort of trace [save for astrally]. But what about mana combat spells like mana bolt?

And how would a spell like Improved invisibility effect the visibility of the spell?

1 Answer
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"Noticing magic is a Simple Perception + Intuition [Mental] Test with
a threshold equal to..." + "Obviously, if a magician is throwing fire
from his fingertips, you’re probably going to notice without making a
test. But if the magic is subtle, then you have to pick up some dice."

p.280-281 "Perceiving Magic", SR5

The same section brings up manabolt as an example. :)

"This spell makes the subject more difficult to detect by normal
visual senses" + "Invisibility affects the minds of viewers. Improved
Invisibility mimics the subject’s surroundings from all angles and
affects technological sensors as well."

p.291, "Improved Invisibility", SR5

I think an attack spell is not part of the "subject" here. Also, I don't think spells have only a visual component.